# Let R be the region enclosed by #f(x) = sinx, g(x) =1-x, and x=0#. What is the volume of the solid produced by revolving R around the x-axis?

##### 1 Answer

Aug 9, 2017

Please see below.

#### Explanation:

Here is a picture of the region with a slice taken perpendicular to the axis of rotation.

Let

The volume of the solid is

# = pi int_0^c ((1-x)^2 - 1/2(1-cos(2x)) dx#

# = pi[ -(1-x)^3/3-1/2x +1/2sinxcosx]_0^c#

# = pi(-(1-c)^3/3+c/2+1/2sin(c) cos(c)+1/3)#

If desired, we can rewrite using

Or we can evaluate using